Drop an ice cube in a glass of warm water. It will melt as you can probably see. It won't melt, no matter how it looks.
This melting process is described by equations by mathematicians. Although the equations are well-tested, it took 130 years to show that they correspond to real facts. Alessio Figalli, JoaquimSerra of the Swiss Federal Institute of Technology Zurich, and Xavier Ros-Oton of University of Barcelona published a paper in March proving that the equations match intuition. Although snowflakes may be possible, they are very rare and fleeting.

Original story reprinted by permission of Quanta Magazine. This independent, editorially-controlled publication of Simons Foundation aims to increase public understanding of science through covering trends and research developments in mathematics and the life and physical sciences.

Maria Colombo, of the Swiss Federal Institute of Technology Lausanne, stated that these results offer a fresh perspective on the subject. This phenomenon was not previously understood in such depth and detail.

The Stefan problem is the question of how ice melts into water. It was posed by Josef Stefan in 1889. This is the best example of a "free boundary problem", where mathematicians examine how a process such as the diffusion of heat can make a boundary move. The boundary in this example is between water and ice.

Mathematicians have been trying to understand these complex models for many years. The new work is inspired by previous research on soap films, which offers a unique type of physical system. They are based on it to show that sharp edges or cusps along the boundary between water and ice rarely form and disappear immediately.

These sharp spots are known as singularities. They are, as it turns out, just as fleeting in the mathematical free boundaries as they are in reality in the physical world.

Melting Hourglasses

Think of an ice cube placed in a glass water. Both substances are composed of the same water molecules but the water molecules are in two distinct phases: liquid and solid. The boundary is where the two phases meet. The boundary is formed when the two phases meet. However, heat from the water melts the ice and the boundary becomes mobile. The boundary and the ice eventually disappear.

The melting boundary will always remain smooth, according to intuition. You don't cut yourself when you take a piece of ice out of a glass of water. It is possible to imagine situations where there are sharp edges if you have a little imagination.

Submerge an hourglass-shaped piece of ice. The ice melts and the hourglass' waist becomes thinner, until the liquid eats through it. This is when a once smooth waist becomes two pointed cusps or singularities.

Giuseppe Mingione, University of Parma, stated that this is one of those problems which naturally displays singularities. That is because of the physical reality.

Josef Stefan developed equations to model melting ice. Archive of the University of Vienna Originator R. Fenzl Signatur 135.726

But reality tells us that singularities can be controlled. Cusps shouldn't last for long because warm water should quickly melt them down. A snowflake could form if you start with an enormous block of ice made entirely from hourglasses. It wouldn't last longer than an instant.

Stefan analyzed the problem mathematically in 1889 and created two equations to describe melting ice. The first describes how heat is absorbed from warm water into cool ice. This shrinks the ice and causes the water region to expand. The melting process is followed by a second equation that tracks the changes in the interface of ice and water. The equations could also be used to describe situations where the ice melts and causes water around it to freeze. However, this possibility is not considered in the current work.